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Chapter 2: Problem 4

a) Solve the equation \(2 \sqrt{3 x+5}+7=16, x \geq-\frac{5}{3}\) algebraically. b) Show how you can use the graph of the function \(y=2 \sqrt{3 x+5}-9, x \geq-\frac{5}{3}\) to find the solution to the equation in part a).

Short Answer

Expert verified
a) \( x = \frac{61}{12} \). b) Find where the graph of \( y = 2 \sqrt{3x+5} - 9 \) intersects y = 7.

Step by step solution

01

Isolate the Square Root Term

Start by subtracting 7 from both sides of the equation to isolate the square root term: \[ 2 \sqrt{3x + 5} + 7 = 16 \] becomes \[ 2 \sqrt{3x + 5} = 9 \]
02

Divide by 2

Next, divide both sides by 2 to further isolate the square root: \[ \sqrt{3x + 5} = \frac{9}{2} \]
03

Square Both Sides

Square both sides of the equation to eliminate the square root: \[ (\sqrt{3x + 5})^2 = \left( \frac{9}{2} \right)^2 \] This simplifies to \[ 3x + 5 = \frac{81}{4} \]
04

Solve for x

Isolate x by first subtracting 5 from both sides: \[ 3x + 5 - 5 = \frac{81}{4} - 5 \] This simplifies to \[ 3x = \frac{81}{4} - \frac{20}{4} = \frac{61}{4} \] Finally, divide both sides by 3: \[ x = \frac{61}{4} \div 3 = \frac{61}{12} \]
05

Verify the Solution

Check that the solution \( x = \frac{61}{12} \) satisfies the original equation and the condition \( x \geq -\frac{5}{3} \).
06

Using the Graph

Rewrite the given function for part b) to match the form in part a): \[ y = 2 \sqrt{3x+5} - 9 \] Set the function equal to 7 (since \(7 + 9 = 16\)): \[ 2 \sqrt{3x+5} - 9 = 7\] Adding 9 to both sides, we get our original equation: \[ 2 \sqrt{3x+5} = 16 \] By plotting the graph of \(y = 2 \sqrt{3x+5} - 9\) and finding the intersection where y = 7, you will find the same x-value solution that solves the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Solution
Solving square root equations algebraically often involves isolating the square root term first. In the given equation, we have:
\[ 2 \sqrt{3x + 5} + 7 = 16 \] Let's isolate the square root term:
Subtract 7 from both sides:
\[ 2 \sqrt{3x + 5} = 9 \] Next, divide by 2:
\[ \sqrt{3x + 5} = \frac{9}{2} \]
To get rid of the square root, you need to square both sides of the equation:
\[ (\sqrt{3x + 5})^2 = \left( \frac{9}{2} \right)^2 \]
This simplifies to:
\[ 3x + 5 = \frac{81}{4} \]
Now, isolate x:
Subtract 5 from both sides:
\[ 3x = \frac{81}{4} - \frac{20}{4} = \frac{61}{4} \]
Lastly, divide by 3:
\[ x = \frac{61}{12} \]
After solving for x, always check your solutions by plugging them back into the original equation. This step is crucial to avoid extraneous solutions that may arise during squaring.
Graphical Solution
Using graphical methods to solve equations provides a visual way to find solutions. From the given equation: \[ 2 \sqrt{3x+5} - 9 = 7 \]
We can rearrange it to align with the original equation:
\[ 2 \sqrt{3x+5} = 16 \]
To solve this graphically, we plot \[ y = 2 \sqrt{3x+5} - 9 \] and look for the intersection where \[ y = 7 \].
Here's how:
  • Plot the function \[ y = 2 \sqrt{3x+5} - 9 \].
  • Draw a horizontal line at \[ y = 7 \].
  • The intersection point of the function and the line gives the x-value that solves the equation.
By mapping the intersection, students can visually confirm the solution they derived algebraically.
Equation Verification
It's essential to verify solutions to avoid mistakes. After finding \[ x = \frac{61}{12} \], substitute this x back into the original equation:
\[ 2 \sqrt{3 \left( \frac{61}{12} \right) + 5} + 7 = 16 \]
Simplify inside the square root:
\[ 2 \sqrt{\frac{183}{12} + 5} + 7 = 16 \]
Convert 5 to a fraction:
\[ 2 \sqrt{\frac{183}{12} + \frac{60}{12}} + 7 = 16 \]
Further simplify:
\[ 2 \sqrt{\frac{243}{12}} + 7 = 16 \]
\[ 2 \sqrt{\frac{81}{4}} + 7 = 16 \]
\[ 2 \left( \frac{9}{2} \right) + 7 = 16 \]
This verifies that \[ x = \frac{61}{12} \] is indeed a valid solution. Checking solutions helps ensure accuracy and reinforces algebraic understanding for students.
Function Transformation
Understanding how functions transform can simplify solving equations. The given function:
\[ y = 2 \sqrt{3x+5} - 9 \]
Includes a square root, a multiplication, and a subtraction.
Here are the key transformations:
  • The term inside the square root \[ 3x+5 \] affects the horizontal shift and scaling.
  • The coefficient 2 scales the function vertically.
  • The subtraction of 9 shifts the function downwards by 9 units.
Recognizing these transformations aids in graphing, simplifying, and solving complex equations. It allows a deeper understanding of how each part affects the graph and ultimately helps find solutions more efficiently.

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Most popular questions from this chapter

Write the equation of the radical function that results by applying each set of transformations to the graph of \(y=\sqrt{x}\). a) vertical stretch by a factor of \(4,\) then horizontal translation of 6 units left b) horizontal stretch by a factor of \(\frac{1}{8},\) then vertical translation of 5 units down c) horizontal reflection in the \(y\) -axis, then horizontal translation of 4 units right and vertical translation of 11 units up d) vertical stretch by a factor of 0.25 vertical reflection in the \(x\) -axis, and horizontal stretch by a factor of 10

a) Identify the domains and ranges of $$y=x^{2}-4 \text { and } y=\sqrt{x^{2}-4}$$. b) Why is \(y=\sqrt{x^{2}-4}\) undefined over an interval? How does this affect the domain of the function?

a) Determine the root(s) of the equation \(\sqrt{x+7}-4=0\) algebraically. b) Determine the \(x\) -intercept(s) of the graph of the function \(y=\sqrt{x+7}-4\) graphically. c) Explain the connection between the root(s) of the equation and the \(x\) -intercept(s) of the graph of the function.

For an observer at a height of \(h\) feet above the surface of Earth, the approximate distance, \(d,\) in miles, to the horizon can be modelled using the radical function \(d=\sqrt{1.50 h}\). a) Use the language of transformations to describe how to obtain the graph from the base square root graph. b) Determine an approximate equivalent function of the form \(d=a \sqrt{h}\) for the function. Which form of the function do you prefer, and why? c) A lifeguard on a tower is looking out over the water with binoculars. How far can she see if her eyes are \(20 \mathrm{ft}\) above the level of the water? Express your answer to the nearest tenth of a mile.

Investigate how the constants in radical functions affect their graphs, domains, and ranges. Step 1 Graph the function \(y=\sqrt{a^{2}-x^{2}}\) for various values of \(a .\) If you use graphing software, you may be able to create sliders that allow you to vary the value of \(a\) and dynamically see the resulting changes in the graph. Step 2 Describe how the value of \(a\) affects the graph of the function and its domain and range. Step 3 Choose one value of \(a\) and write an equation for the reflection of this function in the \(x\) -axis. Graph both functions and describe the graph. Step 4 Repeat steps 1 to 3 for the function \(y=\sqrt{a^{2}+x^{2}}\) as well as another square root of a function involving \(x^{2}\)
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